If I get a call on Skype my phone doesn't ring. I only get the call when I'm on the app.

Same here. I tried signing in both with Skype username and Microsoft account, nothing changed.
Moreover toast notifications don't always work fine: I just received a notification about a message sent to me a couple of days ago.

My experience varies; it will work great in the background for a couple days, notifying me of calls and messages, then it will stop and no longer work at all for a few days. I really hope they can get this working consistently... the last update helped. Before that I never got toasts or call notifications.